Modern Control System Theory and Design, 2nd Edition

8.3.  CONTROLLER DESIGN USING POLE PLACEMENT AND LINEAR-STATE-VARIABLE FEEDBACK TECHNIQUES

The preceding section has indicated several important relationships between open-loop and closed-loop transfer functions. This is very important in the design of control systems for the case where the closed-loop transfer function is specified and it is desired to determine the open-loop transfer function. A typical problem might specify the desired velocity constant; then use is made of Eq. (5.35) in Section 5.4 which gave the velocity constant in terms of the closed-loop poles and zeros. The problem is to determine the resulting linear-state-variable feedback system.

Let us illustrate the procedure by considering the following problem. It is desired that the closed-loop characteristics of a unity-feedback control system be given by the following parameters:

ω_n = 50 rad/sec,      K_v = 35/sec,      ζ = 0.707

What form of closed-loop transfer function will satisfy these requirements? Let us first try a simple quadratic control system having a pair of complex-conjugate poles. From Eq. (5.37), such a system has a velocity constant given by

Therefore, a simple quadratic control system having a pair of complex-conjugate poles will satisfy these specifications. From Eq. (4.18),

For a damping ratio ...